Traditional Numerical Methods Research Articles

Design optimization of groundwater circulation well based on numerical simulation and machine learning

The optimal design of groundwater circulation wells (GCWs) is challenging. The key to purifying groundwater using this technique is its proficiency and productivity. However, traditional numerical simulation methods are limited by long modeling times, random optimization schemes, and optimization results that are not comprehensive. To address these issues, this study introduced an innovative approach for the optimal design of a GCW using machine learning methods. The FloPy package was used to create and implement the MODFLOW and MODPATH models. Subsequently, the formulated models were employed to calculate the characteristic indicators of the effectiveness of the GCW operation, including the radius of influence (R) and the ratio of particle recovery (Pr). A detailed collection of 3000 datasets, including measures of operational efficiency and key elements in machine learning, was meticulously compiled into documents through model execution. The optimization models were trained and evaluated using multiple linear regression (MLR), artificial neural networks (ANN), and support vector machines (SVM). The models produced by the three approaches exhibited notable correlations between anticipated outcomes and datasets. For the optimal design of circulating well parameters, machine learning methods not only improve the optimization speed, but also expand the scope of parameter optimization. Consequently, these models were applied to optimize the configuration of the GCW at a site in Xi’an. The optimal scheme for R (Q = 293.17 m3/d, a = 6.09 m, L = 7.28 m) and optimal scheme for Pr (Q = 300 m3/d, a = 3.64 m, L = 1 m) were obtained. The combination of numerical simulations and machine learning is an effective tool for optimizing and predicting the GCW remediation effect.

High-Speed Surface Property Recognition with a 140 GHz Frequency

In the field of integrated sensing and communication, there is a growing need for advanced environmental perception. The terahertz (THz) frequency band, significant for ultra-high-speed data connections, shows promise in environmental sensing, particularly in detecting surface textures crucial for autonomous systems’ decision-making. However, traditional numerical methods for parameter estimation in these environments struggle with accuracy, speed, and stability, especially in high-speed scenarios like vehicle-to-everything communications. This study introduces a deep learning approach for identifying surface roughness using a 140-GHz setup tailored for such conditions. A high-speed data acquisition system was developed to mimic real-world scenarios, and a diverse set of rough surface samples was prepared for realistic high-speed datasets to train the models. The model was trained and validated in three challenging scenarios: random occlusions, sparse data, and narrow-angle observations. The results demonstrate the method’s effectiveness in high-speed conditions, suggesting terahertz frequencies’ potential in future sensing and communication applications.

AN EFFICIENT SPECTRAL METHOD FOR NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

For Volterra integro-differential equations (VIDEs) with weakly singular kernels, their solutions are singular at the initial time. This property brings a great challenge to traditional numerical methods. Here, we investigate the numerical approximation for the solution of the nonlinear model with weakly singular kernels. Due to its characteristic, we split the interval and focus on the first one to save operation. We employ the corresponding singular functions as basis functions in the first interval to simulate its singular behavior, and take the Legendre polynomials as basis functions in the other one. Then the corresponding hp-version spectral method is proposed, the existence and uniqueness of solution to the numerical scheme are proved, the hp-version optimal convergence is derived. Numerical experiments verify the effectiveness of the proposed method.

Weak-formulated physics-informed modeling and optimization for heterogeneous digital materials

Abstract Numerical solutions to partial differential equations (PDEs) are instrumental for material structural design where extensive data screening is needed. However, traditional numerical methods demand significant computational resources, highlighting the need for innovative optimization algorithms to streamline design exploration. Direct gradient-based optimization algorithms, while effective, rely on design initialization and require complex, problem-specific sensitivity derivations. The advent of machine learning offers a promising alternative to handling large parameter spaces. To further mitigate data dependency, researchers have developed physics-informed neural networks (PINNs) to learn directly from PDEs. However, the intrinsic continuity requirement of PINNs restricts their application in structural mechanics problems, especially for composite materials. Our work addresses this discontinuity issue by substituting the PDE residual with a weak formulation in the physics-informed training process. The proposed approach is exemplified in modeling digital materials, which are mathematical representations of complex composites that possess extreme structural discontinuity. This paper also introduces an interactive process that integrates physics-informed loss with design objectives, eliminating the need for pre-trained surrogate models or analytical sensitivity derivations. The results demonstrate that our approach can preserve the physical accuracy in data-free material surrogate modeling but also accelerates the direct optimization process without model pre-training.

Performance degradation model and life prediction method of spool pair caused by erosion wear

The erosion of solid particles in the fluid causes the wear of the metering edge of the spool pair, resulting in a decrease in system performance. This paper aims to develop a general method to evaluate performance degradation and life prediction of spool pairs induced by erosion. Considering the diversity of influencing factors and the complexity of operating conditions, traditional numerical calculation methods will consume huge computational costs. It is difficult to qualitatively and quantitatively analyze the evolution of metering edge by numerical calculation. The erosion prediction model (EPM) constructed by machine learning methods can be used to predict the wear profile discretized in time and space. Combined with the mathematical model of the valve-controlled cylinder, a performance degradation model (PDM) that can simulate real-time working conditions is established. Through the data exchange and iterative update between EPM and PDM, a complete digital platform for the calculation of slide valve durability is constructed. After comparing with the experimental results, it can be concluded that the proposed method can realize the performance degradation evaluation and life prediction of the spool pair caused by erosion wear under continuous working conditions.

Evolution of mechanical properties of shield tunnels induced by water-soil gushing

Water-soil gushing in shield tunnels has been a major threat to the safety of tunnel structures. However, traditional numerical methods cannot reasonably simulate large soil deformation around the tunnel caused by the water-soil gushing. Therefore, it is difficult to reveal the evolution of the internal force and deformation of tunnel structures. In this paper, based on the two-phase Material Point Method, the numerical analysis method for the water-soil gushing was established to investigate the development of water-soil pressure acting on the tunnel surface, and then the evolution of the tunnel mechanical behaviours in the process of tunnel gushing was further studied. The results indicate that during the “rapid gushing stage”, the tunnel surface pressures around the leakage point decrease sharply, whereas those on the tunnel invert increase significantly. This gives rise to a significant increase in the shear force and bending moment of the tunnel lining, with increase rates of 114% and 82%, respectively. In addition, the axial and angular deformation of Joint No.2 near the leakage point increased significantly, which may lead to serious longitudinal joint dislocation or even concrete crushing. Thus, close attention to the joint deformation is required.

An industrial IoT-based deformation resistance prediction and thickness control method of cold-rolled strip in steel production systems

Different from traditional analytical and numerical simulation methods, data-driven methods can more effectively describe the effects of nonlinear and coupling factors. Although the fluctuations of deformation resistance and thickness for hot-rolled strips significantly influence the thickness accuracy in strip cold rolling, the traditional deformation resistance prediction model and thickness control mode don't involve that. Thus, based on the Industrial Internet of Things (IIOT) platform and data-driven technology, this work proposed a new method for deformation resistance prediction and a new mode of feed-forward control for strip thickness. IIOT platform can collect production data from the various levels to provide a complete data set. The deformation resistance prediction model is established by adopting the differential evolutionary algorithm to optimize the bi-directional long and short-term memory network. In addition, the feed-forward control (DFFC) strategy for strip thickness is proposed based on deformation resistance prediction. An application results of a 1420 mm production line of CR indicated that the established prediction model could provide a high accuracy prediction for deformation resistance, and compared to traditional thickness gauges-based automatic gauge control methods, the DDFC can capture the effect of the fluctuations of deformation resistance and thickness for the hot-rolled strip to improve thickness accuracy effectively.

Simulation of thermal-fluid coupling in silicon single crystal growth based on gradient normalized physics-informed neural network

The thermal-fluid coupling phenomenon of silicon melt is significant in the growth process of silicon single crystals. Complex convection affects the temperature and concentration distribution of the silicon melt. Therefore, establishing and solving the thermal-fluid coupling model of silicon melts is crucial to optimizing the crystal growth process and improving crystal quality. Traditional numerical simulation methods have limitations in regard to optimization, control, and real-time monitoring. Physics-Informed Neural Network (PINN) does not require model discretization, after training, it can make predictions quickly, showing potential for industrial applications. However, when solving practical industrial coupling models, PINN often struggles to converge due to large parameter values and significant gaps between solution variables. Moreover, solving the thermal-fluid coupling model with PINN can be treated as a multitask problem, where the gradients of different equations interfere with each other, leading to gradient confusion, slow convergence, or even divergence. Therefore, this paper proposes a gradient normalized PINN (GNPINN) for solving the thermal-fluid coupling model of silicon melt. GNPINN balances the contribution of each task, ensuring a more equitable training speed between different tasks to stabilize the training process of the coupling model. This paper considers the thermal-fluid coupling model of silicon melt under different rotation conditions. GNPINN can accurately and comprehensively capture the complex temperature, velocity, and pressure distribution of silicon melt compared with other methods. Additionally, the experimental results uncover the flow and heat transfer properties of silicon melt, validating the effectiveness and industrial applicability of GNPINN.

DualFluidNet: An attention-based dual-pipeline network for fluid simulation

Fluid motion can be considered as a point cloud transformation when using the SPH method. Compared to traditional numerical analysis methods, using machine learning techniques to learn physics simulations can achieve near-accurate results, while significantly increasing efficiency. In this paper, we propose an innovative approach for 3D fluid simulations utilizing an Attention-based Dual-pipeline Network, which employs a dual-pipeline architecture, seamlessly integrated with an Attention-based Feature Fusion Module. Unlike previous methods, which often make difficult trade-offs between global fluid control and physical law constraints, we find a way to achieve a better balance between these two crucial aspects with a well-designed dual-pipeline approach. Additionally, we design a Type-aware Input Module to adaptively recognize particles of different types and perform feature fusion afterward, such that fluid-solid coupling issues can be better dealt with. Furthermore, we propose a new dataset, Tank3D, to further explore the network’s ability to handle more complicated scenes. The experiments demonstrate that our approach not only attains a quantitative enhancement in various metrics, surpassing the state-of-the-art methods, but also signifies a qualitative leap in neural network-based simulation by faithfully adhering to the physical laws. Code and video demonstrations are available at https://github.com/chenyu-xjtu/DualFluidNet.

Numerical solutions of boundary problems in partial differential equations: A deep learning framework with Green's function

This work introduces Green Networks (GreenNets), a novel class of neural network architectures designed to address a wide spectrum of boundary problems in partial differential equations (PDEs). Demonstrated versatility includes solving the Poisson equation, convection-diffusion equation, and unsteady incompressible Navier-Stokes equations. GreenNets comprises two specialized variants: Boundary Transformation Green Network (BT-GreenNet), focusing on transforming Dirichlet boundary conditions into source terms, and Boundary Integral Green Network (BI-GreenNet), considering boundary integral terms with shared weights between the boundary and source integrals. Comparisons with existing deep learning and traditional numerical methods underscore their accuracy and efficiency. BT-GreenNet excels in efficiency, and BI-GreenNet offers enhanced flexibility. The research pinnacle, Boundary Unified Green Network (BU-GreenNet), merges BT-GreenNet and BI-GreenNet to address coupled PDEs, specifically unsteady incompressible Navier-Stokes equations. BU-GreenNet stands out by surpassing traditional numerical approaches, offering remarkable computational efficiency while faithfully capturing the intricate evolution of fluid dynamics phenomena. This research redefines the means of solution to boundary problems in PDEs through innovative neural network architectures, emphasizing practical utility and computational efficiency.

Physics-informed neural network for diffusive wave model

The diffusive wave model (DWM), a nonlinear second-order simplified form of the shallow water equation, has been widely used in hydraulic, hydrologic and irrigation engineering. Solving the forward problem of the DWM can be utilized to predict evolution in water levels and discharge. Solving its inverse problem allows for the identification of crucial parameters (such as Manning's coefficient, rainfall intensity, etc.) based on observations. This paper applies the physics-informed neural network (PINN) with novel improvements to solve the DWM for both forward and inverse problems. In the forward problem, compared to traditional numerical methods, PINN can predict the evolution at any location. In the inverse problem, PINN provides a simple and efficient solution process. In order to overcome the gradient explosion in the training process caused by the characteristics of the DWM, the stop-gradient technique was adopted to train the neural network. To improve the estimation of DWM parameters, the concept of time division was developed, and a new network structure was then proposed. To verify the effectiveness of PINN and its improved algorithm for DWM, seven examples were simulated. The PINN solutions for forward problems were compared with the results obtained by classical numerical methods, while the correct rainfall pattern was learnt for the inverse problem.

Thermo-magnetic radiative flow in porous enclosure with deep-learning parameter estimation

This study numerically investigates convective heat transfer, fluid flow, and entropy generation within an inverted T-shaped porous enclosure filled with a hybrid nanofluid. The interplay between thermal radiation and a magnetic field is considered, as these factors significantly influence thermal acquisition processes in renewable and thermal engineering applications, particularly solar power collectors with the selected physical domain. By analyzing various combinations of flow parameters, this work aims to identify favourable conditions for maximizing heat transfer efficiency in such systems. A Darcy-Forchheimer-Brinkman-based mathematical model has been incorporated and numerically simulated using the penalty finite element method (FEM). Moreover, the variation of fluid flow, isotherms, and entropy generation is thoroughly analyzed for the variation of non-dimensional permeability parameter (Darcy number, Da), buoyancy force (Rayleigh number, Ra), magnetic field (Hartmann number, Ha), porosity value (ϵ), and thermal radiation parameter (Rd). The comprehensive result demonstrates that increasing values of Ra, Da, ϵ, and Rd intensify entropy generation, convective heat, and fluid flow phenomena, whereas the increasing value of Ha substantially weakens these phenomena. The thermal efficiency of the model attends at the lower buoyancy forces (Ra≤105) for each flow parameter (except for Ha). However, the increasing magnetic forces (Ha) substantially improve the thermal efficiency of the developed model. Additionally, this study employs a feed-forward neural network (FNN) based deep learning (DL) approach to predict the parametric influence on convective heat transport rate (Num), total entropy generation rate (Stot), and total Bejan number due to heat transport irreversibility (Beθ,tot) based on the minimally supplied experimental data for FNN training process. The trained FNN model predicts the results very quickly with 98.6% accuracy compared to FEM results, demonstrating its efficiency in approximating the parametric influence of complex mathematical model results. Moreover, this approach significantly reduces computational costs and efforts compared to traditional numerical methods, making it a valuable tool for forecasting the influence of an unknown set of flow parameters in complex mathematical models.

Reconstruction of internal heat source in biological tissue using parallel particle swarm optimization

With the rapid development of engineering thermophysics, researches on human biological heat transfer phenomena has gradually shifted from qualitative to quantitative. The heat distribution information of human lesions is of great value for disease analysis, diagnosis, and treatment. It is a typical inverse problem of heat conduction that deriving the distribution of internal heat sources from the temperature distribution on the body surface. Differing from traditional numerical methods for solving heat conduction, this paper transforms such an inverse problem of bio-heat transfer into a direct one, thereby avoiding complex boundary conditions and regularization processes. To noninvasively reconstruct the internal heat source and its corresponding 3D temperature field in biological tissue, the parallel particle swarm optimization (PPSO) is used in the simulation module, where the position P(x, y, z) of point heat source in biological tissue and its corresponding temperature T are set as the optimization variables. Under a certain optimized sample, one can obtain the simulated temperature distributing on the surface of the module, then subtract the simulated temperature from the measured temperature of the same surface which was measured using a thermal infrared imager. If the absolute value of the difference is smaller, it indicates that the current sample is closer to the true location and temperature of the heat source. When the values of optimization variables are determined, the corresponding 3D temperature field is also confirmed. The simulation results show the simulated position and temperature of the heat source are very approximate with those of the real experimental module. The method presented in this paper has enormous potential and promising prospects in clinical research and application, such as tumor hyperthermia, disease thermal diagnosis technology, etc.

Effective Elastic Properties and Micro-mechanical Damage Evolution of Composite Granular Rocks: Insights from Particulate Discrete Element Modelling

AbstractThe mechanical behavior of composite granular rocks is a multifaceted phenomenon with broad relevance in various geomechanical applications. Traditional homogenization models and continuum mechanics-based numerical methods often fall short of accurately capturing the intricacies of granular materials. Granular materials exhibit heterogeneity and arching mechanisms governing the force networks that ensure system stability. Unlike continuum-based approaches, discrete element methods (DEM) have an advantage in assessing effective material properties by considering material heterogeneity and grain-level physical interactions. This study evaluates effective elastic properties using DEM with flat-joint contact law for composite binary mixtures with a stiff inclusion embedded within a matrix material. We examine variations in the elastic properties across different structural and laminated geometrical distributions of inclusion materials. Our findings closely adhere to the Voigt-Reuss and Hashin–Strikman bounds within their specific conditions, demonstrating the promising application of DEM in the analysis of composite materials. In addition, our research provides an in-depth analysis of the distinctive stress-evolution and damage-evolution characteristics exhibited by various geometrical configurations of inclusions under unconfined compressive loading. These results offer invaluable insights into the mechanical behavior of composite granular rocks and underscore the potential applications of DEM in addressing rock physics modeling problems encountered in petroleum engineering.

Mathematical modelling of HMT through porous stretching sheet using artificial neural network

The influence of heat radiation on HMT, MHD on a porous stretched sheet has been explained in this study. The governing PDEs are converted to ODEs via similarity transformations. The coefficient of skin friction, rate of HMT are calculated using the MATLAB software for various parameter values. The controlling boundary layer PDEs are turned into a system of coupled nonlinear ODEs via similarity techniques, which are numerically solved using the shooting technique in conjunction with the fourth order Runge Kutta method, and then ANN is applied to them. Validation of numerical results using ANN results. We discovered that using engineering standpoints, an ANN model may produce high-efficiency estimates for heat transfer rates. The achieved R squared values of 99% for predicting Skin friction, Nusselt number and Sherwood number coefficients highlight the remarkable effectiveness of neural network models in these predictions. This efficacy not only demonstrates the accuracy of the models but also results in a significant reduction in the computational time required compared to traditional numerical methods. Furthermore, when compared to alternative numerical approaches, the current ANN model stands out for its applicability to more complex mathematical models, because its efficiency in minimizing both time and processing capacity demands in solving such problems.

SR-M−GAN: A generative model for high-fidelity stress fields prediction of the composite bolted joints

Predicting the stress fields induced by composite bolted joints constitutes a valuable research endeavor aimed at enhancing the structural integrity of equipment in the aerospace and maritime fields. The stress field characteristics of composite bolted joints may undergo changes due to potential mechanical behaviors, such as damage, constitutive relationships and so on, which exhibit complexity and pose challenges for prediction. Nevertheless, modeling the intricate stress field distribution directly from experimental data poses a formidable challenge, necessitating a substantial investment in materials and resources. Traditional finite element numerical methods have the capability to model the distribution of the stress fields. However, to obtain a high-fidelity solution of this distribution, it is imperative to generate a fine-grid mesh and allocate the significant computational resources. The incorporation of computational techniques in coarse −grid would enhance the efficiency of computations, but these approaches compromise the accuracy of the calculations. In this paper, an innovative deep learning framework is introduced. The network named as the SR-M−GAN is drew inspiration from the conditional generative adversarial network (cGAN) used in the image super-resolution. The primary aim of SR-M−GAN is to mitigate the intrinsic trade-off between result fidelity and computational complexity. The model proposed in this paper effectively reconstructs high-fidelity stress field data from calculations on coarse grids and exhibits excellent reconstruction metrics. Simultaneously, the model only incurs additional computational costs during the deployment of the network model and training. When compared to experimental data, the model continues to demonstrate superior performance in both accuracy and computational efficiency. Therefore, this method offers a high-fidelity data augmentation approach for stress analysis in composite bolted joints, laying the groundwork for large-scale analysis of stress field characteristics in composite bolted joints.

A transformer-based method for correcting significant wave height numerical forecasting errors

Accurate significant wave height (SWH) forecasting is essential for various marine activities. While traditional numerical and mathematical-statistical methods have made progress, there is still room for improvement. This study introduces a novel transformer-based approach called the 2D-Geoformer to enhance SWH forecasting accuracy. The 2D-Geoformer combines the spatial distribution capturing capabilities of SWH numerical models with the ability of mathematical-statistical methods to identify intrinsic relationships among datasets. Using a comprehensive long time series of SWH numerical hindcast datasets as the numerical forecasting database and ERA5 reanalysis SWH datasets as the observational proxies database, with a focus on a 72-hour forecasting window, the 2D-Geoformer is designed. By training the potential connections between SWH numerical forecasting fields and forecasting errors, we can retrieve SWH forecasting errors for each numerical forecasting case. The corrected forecasting results can be obtained by subtracting the retrieved SWH forecasting errors from the original numerical forecasting fields. During long-term validation periods, this method consistently and effectively corrects numerical forecasting errors for almost every case, resulting in a significant reduction in root mean square error compared to the original numerical forecasting fields. Further analysis reveals that this method is particularly effective for numerical forecasting fields with higher errors compared to those with relatively smaller errors. This integrated approach represents a substantial advancement in SWH forecasting, with the potential to improve the accuracy of operational SWH forecasts. The 2D-Geoformer combines the strengths of numerical models and mathematical-statistical methods, enabling better capture of spatial distributions and intrinsic relationships in the data. The method's effectiveness in correcting numerical forecasting errors, particularly for cases with higher errors, highlights its potential for enhancing SWH forecasting accuracy in operational settings.

Mathematical insights of social media addiction: fractal-fractional perspectives

The excessive use of social media has become a growing concern in the current century, with dependence on these platforms developing into a complex behavioral addiction. Addressing this issue requires the employment of well-directed and inclusive efforts. In pursuit of continuous development in existent strategies, this article presents a non-linear deterministic mathematical model that encapsulates the dynamics of social media addiction within a population. The proposed model incorporates the fractal-fractional order derivative in the sense of the Caputo operator. The objectives of this research are attained by groping the dynamics of the social media addiction model through the stratification of the population into five compartments: susceptible individuals, exposed individuals, addicted individuals, recovery individuals, and those who have quit using social media. The validity of the devised model is established by proving the existence and uniqueness of the solution within the framework of the fixed-point theory. The Ulam-Hyer’s stability is established through nonlinear functional analysis, perturbing the problem with a small factor. Utilizing the Adam Bashforth numerical scheme, we obtain numerical solutions, which we validate through MATLAB simulations. Additionally, we explore the application of artificial neural networks (ANNs) to approximate solutions, presenting a significant innovation in this domain. We propose the adoption of this novel method for solving integral equations that elucidate the dynamics of social media addiction, surpassing traditional numerical methods. Numerical results are illustrated across various fractional orders and fractal dimensions, with comparisons made against integer orders. Our study indicates that ANN outperforms the Adams-Bashforth algorithm, offering a dependable approach to problem-solving. Throughout the article, we underscore the competitive advantage of our proposed strategy, providing a more nuanced understanding of the complex dynamics outlined in the model.

A fast prediction model of blade flutter in turbomachinery based on graph convolutional neural network

With the development of gas turbine towards higher load capacity and improved efficiency, blade flutter has emerged as a significant obstacle in the further development of turbomachinery due to fluid-structure coupling phenomenon. However, the traditional numerical simulation methods lengthen the design cycle and reduce the optimization efficiency due to the expensive time and resource cost. To solve this problem, a fast prediction model of blade flutter based on the graph convolutional neural network is proposed. The blade flutter prediction graph convolutional neural network containing two sub-networks. The flow field prediction network is used to realize prediction from design variables to flow field parameters. The flutter parameter identification network is adopted to realize identification from flow field parameters to 15 flutter parameters. The prediction performance for physical fields, modal forces, aerodynamic damping and aeroelastic stability are discussed. The influence of training size is explored. The results show that the network can not only accurately capture the trends and details of the physical field, but also obtain the accurate modal force curve, so as to calculate the aerodynamic damping parameters and finally judge the aeroelastic stability.

A linearly implicit energy-stable scheme for critical dissipative surface quasi-geostrophic flows

In this paper, we propose an effective linearly implicit unconditional energy-stable scheme for surface quasi-geostrophic flows based on the scalar auxiliary variable approach and the Fourier spectral Galerkin method. Compared with traditional numerical methods, our scheme has constant coefficient matrices at each time step, and the numerical solutions are consistent with the dissipation laws for modified energy. By treating linear terms implicitly and nonlinear terms explicitly, we derive the dissipation laws for discrete modified surface kinetic energy and Hamiltonian. To reduce the aliasing error induced by the Fourier spectral Galerkin method, we implement a 2/3 de-aliasing technique for the nonlinear terms. Furthermore, the integration concerning energy in our numerical scheme is exact due to the Fourier spectral Galerkin method. Numerical experiments are presented to verify the stability and efficiency of the proposed scheme.